Interaction of Thermal Explosion and Natural Convection: Critical Conditions and New Oscillating Regimes

International audienceIn this paper we investigate, numerically as well as analytically, the influence of natural convection on thermal explosion in a two-dimensional square vessel, filled with a reactant mixture, whose vertical walls are adiabatic and horizontal walls are infinitely conducting, preset at an equal temperature T0. Natural convection enhances the heat losses at the boundaries while large temperatures tend to promote natural convection, thus yielding two competitive phenomena. The governing equations are taken to be the Navier--Stokes equations in the Oberbeck--Boussinesq approximation of low density variations coupled to the heat equation with an exponential chemical source term. This is valid because we consider a 1-step reaction with high heat release, we use the Frank-Kamenetskii transformation under high activation energy asymptotics, and we do not take into account thermo-diffusion as well as the different molar masses of the species. We solve the vorticity-stream function-temperature formulation with an alternating direction numerical method on finite difference approximations. The numerical results show the coexistence of 1-vortex and 2-vortex regimes, from which thermal explosion can occur. New regimes of thermal explosion are found when the Frank-Kamenetskii and Rayleigh parameters are close to the critical conditions for explosion and convection. Periodic-in-time solutions can exist from which thermal explosion can also occur. Linear stability analysis allows us to predict the onset of convection not too close to the explosion limit. Finally, we propose a model problem through a system of two ordinary differential equations which is able to reproduce the bifurcation behavior of the global system close to critical conditions for both explosion and convection